Hausdorff metric / Munkres weirdness
Mar. 20th, 2007 09:07 pmMunkres's Topology, Section 45, exercise 8; we have metric spaces X and Y, the product space X×Y under the max metric; and H, the space of all nonempty, closed, bounded subsets of X×Y under the Hausdorff metric. So Munkres tells us to consider the function gr:C(X,Y)→H (where C(X,Y) is considered in the uniform metric, for the record), where gr assigns to each function its graph, and then...
...well, "and then" doesn't actually matter for now, because gr isn't actually a function from C(X,Y) to H unless we impose some more conditions. For f∈C(X,Y), gr(f) is always closed, but is bounded iff X and f(X) are. So we'll have to require X bounded, but that's not nearly enough to force all continuous functions from X to Y to be bounded... we could limit ourselves to X compact, but that's clearly not what Munkres intended as part (c) of the problem requires a continuous f:X→Y that's not uniformly continuous. We could limit ourselves to Y bounded, but I suspect that if that were the intention, the book would just say "nonempty closed subsets" rather than "nonempty closed bounded subsets". Maybe it's supposed to be BC(X,Y)? Maybe we're just supposed to assume every f∈C(X,Y) is bounded? Maybe there was supposed to be some additional condition which would ensure this? Maybe we were supposed to take the standard bounded metrics on X,Y obtained from dX, dY? Or just on the product space? [And of course in addition to all this we have to assume X bounded if that's not already implied.]
Looking up the errata for Munkres, there are no errata for this page (page 281) (or any nearby pages, if it had moved by a page in other printings). Anyone have any idea what might have been meant here?
-Harry
...well, "and then" doesn't actually matter for now, because gr isn't actually a function from C(X,Y) to H unless we impose some more conditions. For f∈C(X,Y), gr(f) is always closed, but is bounded iff X and f(X) are. So we'll have to require X bounded, but that's not nearly enough to force all continuous functions from X to Y to be bounded... we could limit ourselves to X compact, but that's clearly not what Munkres intended as part (c) of the problem requires a continuous f:X→Y that's not uniformly continuous. We could limit ourselves to Y bounded, but I suspect that if that were the intention, the book would just say "nonempty closed subsets" rather than "nonempty closed bounded subsets". Maybe it's supposed to be BC(X,Y)? Maybe we're just supposed to assume every f∈C(X,Y) is bounded? Maybe there was supposed to be some additional condition which would ensure this? Maybe we were supposed to take the standard bounded metrics on X,Y obtained from dX, dY? Or just on the product space? [And of course in addition to all this we have to assume X bounded if that's not already implied.]
Looking up the errata for Munkres, there are no errata for this page (page 281) (or any nearby pages, if it had moved by a page in other printings). Anyone have any idea what might have been meant here?
-Harry